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On the automorphism group of some pro-l fundamental groups. (English) Zbl 0657.20028
Galois representations and arithmetic algebraic geometry, Proc. Symp., Kyoto/Jap. 1985 and Tokyo/Jap. 1986, Adv. Stud. Pure Math. 12, 137-159 (1987).
[For the entire collection see Zbl 0632.00004.]
Let l be a fixed prime number and \(g\geq 2\) be an integer. Let G be the pro-l completion of the fundamental group of a compact Riemann surface of genus g, i.e. G is a pro-l group generated by 2g elements \(x_ 1,...,x_{2g}\) with one defining relation; \[ [x_ 1,x_{g+1}][x_ 2,x_{g+2}],...,[x_ g,x_{2g}]=1\quad G=<x_ 1,x_ 2,...,x_{2g}| \quad [x_ 1,x_{g+1}],...,[x_ g,x_{2g}]=1>_{pro-l}. \] Let \({\tilde \Gamma}{}_ g\) denote the group of continuous automorphisms of G and \(\Gamma_ g\) denote the outer automorphism group of G; \(\Gamma_ g={\tilde \Gamma}_ g/Int G\), Int G being the inner automorphism group of G. (Note that every continuous automorphism of G is bi-continuous, as G is compact.) Our aim in this paper is to study these groups \({\tilde \Gamma}{}_ g\) and \(\Gamma_ g\). We shall give filtrations of \({\tilde \Gamma}{}_ g\) and \(\Gamma_ g\) and prove a result on conjugacy classes of \(\Gamma_ g\).

20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20E18 Limits, profinite groups
30F10 Compact Riemann surfaces and uniformization
20F28 Automorphism groups of groups